A remark on an integral structure of the imperfect coefficient ring of $(\varphi,\Gamma)$-modules

TL;DR

This paper proves an isomorphism involving the imperfect coefficient ring of $(,)$-modules over a complete discrete valuation field, extending a remark by Wach to ramified cases.

Contribution

It establishes the isomorphism of the canonical map for the imperfect coefficient ring, confirming a previously unproven remark by Wach in ramified settings.

Findings

Proves the isomorphism of $W(k_{K_infty})[[6]]$ and $_K \u2229 A_inf$

Extends Wach's remark to ramified fields

Confirms the structure of the imperfect coefficient ring in general cases

Abstract

Let $K$ be a complete discrete valuation field of characteristic $0$ with perfect residue field of characteristic $p>0$. Let $\mathbb{A}_K$ denote the imperfect coefficient ring of $(\varphi,\Gamma)$-modules defined by Jean-Marc Fontaine. We prove that the canonical map $W(k_{K_\infty})[[\mu]]\rightarrow \mathbb{A}_K\cap A_\mathrm{inf}$ is an isomorphism, even if $K$ is ramified. This fact was remarked by Nathalie Wach without proof.